Paper deep dive
Demographically-Informed Heat-Mortality Risk Curves via Risk Graph Neural Networks
Alex O. Davies, Eunice Lo, Rui Zhu
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 7/24/2026, 2:41:04 AM
Summary
The paper introduces Risk Graph Neural Networks (RGNNs), a hierarchical GNN encoder that augments Distributed Lag Non-linear Models (DLNMs) with granular census features to improve heat-mortality risk estimation. Evaluated across 10 regions in England and Wales, RGNN variants demonstrate superior predictive calibration and uncertainty coverage during extreme heat events (2022 heatwave) compared to traditional DLNMs and other ML baselines, while preserving the interpretability of risk curves.
Entities (10)
Relation Signals (7)
Risk Graph Neural Networks → evaluatedin → England and Wales
confidence 95% · Evaluated across 10 regions of England and Wales
Risk Graph Neural Networks → improves → Distributed Lag Non-linear Models
confidence 95% · RGNN variants maintain both lower point-errors and near-nominal uncertainty coverage during the 2022 heatwave where baselines collapse.
London → isregionof → England and Wales
confidence 90% · Risk surfaces, in units of deaths per 100k, for London.
Risk Graph Neural Networks → outperforms → Distributed Lag Non-linear Models
confidence 90% · In 2022, however, the DLNM’s coverage collapses to well below the nominal 90% level—while the RGNN variants maintain coverage between 0.83 and 0.89.
Risk Graph Neural Networks → uses → Census features
confidence 90% · uses granular census features to optimise DLNM coefficient vectors
Risk Graph Neural Networks → employs → GraphSAGE
confidence 85% · geographic neighbours at the same level are smoothed using SAGEConv mean aggregation
Risk Graph Neural Networks → employs → GATv2
confidence 85% · child embeddings are aggregated into parent nodes using multi-head GATv2Conv attention
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Estimating heat-related mortality risk is a core task in environmental epidemiology, typically addressed with Distributed Lag Non-linear Models (DLNMs); interpretable exposure-response surfaces fitted to temperature-mortality time series. DLNMs are effective but ignore demographic and geographic context, despite well-established relevance to heat vulnerability. We propose Risk Graph Neural Networks (RGNNs), a hierarchical GNN encoder that uses granular census features to optimise DLNM coefficient vectors, preserving interpretable risk curve outputs while substantially improving predictive calibration. Evaluated across 10 regions of England and Wales on two unprecedented heat years, RGNN variants maintain both lower point-errors and near-nominal uncertainty coverage during the 2022 heatwave where baselines collapse.
Tags
Links
- Source: https://arxiv.org/abs/2607.21131v1
- Canonical: https://arxiv.org/abs/2607.21131v1
Trouble viewing inline? Open PDF directly →
Full Text
26,811 characters extracted from source content.
Expand or collapse full text
Demographically-Informed Heat-Mortality Risk Curves via Risk Graph Neural Networks Alex O. Davies 1 (), Eunice Lo 1 , and Rui Zhu 1 School of Geographical Sciences, University of Bristol, UK alexander.davies@bristol.ac.uk Abstract. Estimating heat-related mortality risk is a core task in envi- ronmental epidemiology, typically addressed with Distributed Lag Non- linear Models (DLNMs); interpretable exposure-response surfaces fitted to temperature-mortality time series. DLNMs are effective but ignore de- mographic and geographic context, despite well-established relevance to heat vulnerability. We propose Risk Graph Neural Networks (RGNNs), a hierarchical GNN encoder that uses granular census features to optimise DLNM coefficient vectors, preserving interpretable risk curve outputs while substantially improving predictive calibration. Evaluated across 10 regions of England and Wales on two unprecedented heat years, RGNN variants maintain both lower point-errors and near-nominal uncertainty coverage during the 2022 heatwave where baselines collapse. Keywords: temperature–mortality modelling· graph neural networks· distributed lag non-linear models· spatial epidemiology 1 Introduction A central task in environmental epidemiology is estimating mortality risk from temperature exposure. The dominant tool is the Distributed Lag Non-linear Model (DLNM) [4], which produces an interpretable exposure-response surface over a lag dimension. This captures, for instance, that cold mortality peaks 7–14 days after exposure while heat mortality concentrates in the immediate 3-day window. DLNMs operate purely on temporal signals, however, making no use of demographic or geographic features despite their established importance in mediating heat vulnerability. Recent ML approaches [1] sometimes improve point estimates but rarely match the interpretability of the DLNM risk curve, and Boudreault et al. [2] note that the DLNM is rarely used as a proper benchmark. A core reason for the wide adoption of DLNMs over other predictive models is the interpretability of their outputs. A risk curve is immediately transparent to technical and non-technical users, making it a powerful tool in informing pol- icy decisions. We introduce Risk Graph Neural Networks (RGNNs), which aug- ment DLNMs with a hierarchical GNN encoder over census features. The result preserves the interpretable risk curve while substantially improving calibration under distributional shift — precisely the regime that matters most for public arXiv:2607.21131v1 [cs.LG] 23 Jul 2026 2Davies et al. health decision-making during extreme heat events. Early counterfactual exper- iments produce spatially-resolved, policy-relevant mortality predictions across urban areas. 2 Background Fig. 1. Risk surfaces, in units of deaths per 100k, for London. Left: Vanilla DLNM. Right: RGNN-Calibrated. Here we briefly summarise the current state of modelling the heat-mortality relationship. DLNMs are proposed and formalised in-detail in Gasparrini et al. [4], and Boudreault et al. [2] review ML approaches. 2.1 Distributed Lag Non-linear Models (DLNMs) A DLNM [4] models a scalar health outcome at location i and time t as a linear combination of a cross-basis matrix that encodes the joint non-linear and lagged relationship between an exposure and the outcome: ˆy it = w i · CB it ,(1) where w i ∈R n cb is a coefficient vector estimated by ordinary least squares and CB it ∈R n cb is the cross-basis row for location i at time t. The primary output is a exposure-outcome effect surface along lag and the exposure variable, which in our case is lag-temperature-risk, as visualised in Figure 1. Fitting one DLNM per location independently produces unstable estimates for small areas with sparse death counts, and provides no mechanism for borrowing strength from neighbouring or demographically similar locations. Heat-Mortality Risk with GNNs3 2.2 Graph Neural Networks (GNNs) Graph Neural Networks [9] generalise neural networks to graph-structured data by iteratively aggregating information from local neighbourhoods. GATv2 [13] extends graph attention networks with a more expressive attention mechanism in which both source and target node features jointly determine attention weights, making it well-suited to heterogeneous hierarchies where child nodes vary sub- stantially in informativeness. GraphSAGE [6] performs mean aggregation over sampled neighbourhoods and is widely used for spatial smoothing on geographic graphs. GNNs have been applied to urban analytics tasks including traffic fore- casting [8] and air quality prediction [3], but their application to epidemiological exposure-response estimation is, to our knowledge, novel. 3 Risk Graph Neural Networks (RGNNs) The RGNN replaces per-location independent OLS with a learned mapping from census features to DLNM coefficient vectors, structured as a two-level hierarchy: a frozen per-region OLS baseline is modulated by GNN-derived corrections at both the region and LAD levels. The output preserves both the interpretability and strong prior of the OLS-initialised DLNM, while improving performance with spatially-shared demographic information, as well as being comparably cheap to run compared to full ML models. An overview of the architecture is given in Figure 2. Fig. 2. RGNN architecture. Census features at the lowest administrative level are pro- jected to a hidden embedding and propagated upward and laterally in separate stages, with heads predict modifying weights (Eqn. 2) over an OLS-initialised DLNM. Initialisation. Before GNN training, per-region OLS coefficients b r are estimated on region-level Pearson mortality residuals and frozen. All GNN heads are zero- initialised, so at the start of training every LAD’s coefficient vector equals its region OLS baseline. 4Davies et al. Encoder. A bottom-up heterogeneous GNN propagates static demographic fea- tures from the finest available administrative unit upward through the hierar- chy. At each level, child embeddings are aggregated into parent nodes using multi-head GATv2Conv attention; geographic neighbours at the same level are smoothed using SAGEConv mean aggregation. Higher-level nodes receive no di- rect demographic features; their representations are built entirely from upward message passing, forcing the model to learn meaningful aggregations. Coefficient prediction. Zero-initialised linear heads at each predictive level map GNN embeddings to corrections inR n cb : w i = b r(i) + clamp b r(i) ⊙ f ls (h i ) + f lsh (h i ), ±δ ,(2) where ⊙ is element-wise multiplication, δ is a hyperparameter controlling the maximum correction magnitude, and the scale and shift heads modulate the shape and location of the baseline curve respectively. Importantly, weights at each level anchor independently to the same frozen baseline rather than being chained, keeping granularities independently interpretable and preventing cas- cading errors. With dropout, and repeat predictions of coefficients, we can form useful uncertainties that can be propagated into the vanilla DLNMs uncertainty estimations. Training objective. The primary loss is Lin’s C [10], which jointly penalises errors in correlation, variance ratio, and mean bias, summed at LAD and region levels with population weighting. Three auxiliary terms are added: (i) a mono- tonicity penalty encouraging physiologically plausible cold and heat tails; (i) a consistency penalty penalising divergence between population-weighted LAD aggregates and region-level predictions; and (i) a population-inverse-weighted L2 delta penalty pulling data-sparse LADs toward their regional baseline. MC- dropout over the coefficient heads propagates uncertainty into the final risk curves. 4 Experiments We collect daily temperature data from HadUK [7] and LAD-level daily mor- tality through a data-sharing agreement with the UK ONS (2000–2024). For demographic features we use OA-level census information from Goodwin and Singleton [5] for 2021, with the implicit assumption that demographic drift is minimal across the study period. [Adjacent_To] edges connect areas sharing a boundary, e.g. (Wales) ↔ (South West); [Contains] edges encode the adminis- trative hierarchy, e.g. (Wales) → (Cardiff). We demonstrate three RGNN variants, which share the same codebase and architecture, differing only in the depth of the administrative hierarchy: rgnn (LAD→Region, 2 levels) rgnn_lsoa (LSOA→LAD→Region, 3 levels) rgnn_oa (OA→LSOA→LAD→Region, 4 levels) As baselines we use the DLNM, Random Heat-Mortality Risk with GNNs5 Table 1. Region-level holdout evaluation across 10 regions of England and Wales. Values are mean ± standard deviation across regions. Bold indicates the best result in each column. Year ModelRMSE ↓MAE ↓Pearson ↑CRPS ↓ Cov@90% ↑ 2018 DLNM0.33 ± 0.07 0.25 ± 0.05 0.32 ± 0.05 0.19 ± 0.04 0.73 ± 0.03 RF0.29 ± 0.06 0.22 ± 0.04 0.45 ± 0.04 0.19 ± 0.04 0.36 ± 0.02 GBM0.28 ± 0.06 0.21 ± 0.04 0.37 ± 0.06 0.16 ± 0.03 0.78 ± 0.03 LSTM † 0.27 ± 0.04 0.21 ± 0.03 0.55 ± 0.10— MLP † 0.29 ± 0.05 0.23 ± 0.04 0.48 ± 0.07— RGNN0.24 ± 0.04 0.19 ± 0.03 0.52 ± 0.08 0.14 ± 0.02 0.97 ± 0.02 RGNN-LSOA 0.24 ± 0.05 0.19 ± 0.04 0.56 ± 0.05 0.14 ± 0.02 0.99 ± 0.01 RGNN-OA 0.23 ± 0.04 0.19 ± 0.03 0.56 ± 0.05 0.14 ± 0.02 0.98 ± 0.01 2022 DLNM0.54 ± 0.09 0.44 ± 0.07 0.49 ± 0.11 0.36 ± 0.05 0.43 ± 0.06 RF0.52 ± 0.08 0.43 ± 0.06 0.46 ± 0.07 0.40 ± 0.06 0.11 ± 0.03 GBM0.52 ± 0.08 0.43 ± 0.06 0.40 ± 0.08 0.35 ± 0.05 0.39 ± 0.07 LSTM † 0.53 ± 0.07 0.46 ± 0.06 0.40 ± 0.13— MLP † 0.53 ± 0.08 0.46 ± 0.07 0.48 ± 0.11— RGNN0.39 ± 0.04 0.31 ± 0.03 0.47 ± 0.07 0.22 ± 0.02 0.87 ± 0.03 RGNN-LSOA 0.39 ± 0.03 0.32 ± 0.02 0.40 ± 0.07 0.22 ± 0.02 0.89 ± 0.03 RGNN-OA 0.43 ± 0.04 0.36 ± 0.03 0.40 ± 0.07 0.25 ± 0.02 0.83 ± 0.04 Forests (RF), Gradient Boosted Models (GBM), Long Short-Term Memory net- works (LSTMs) and Multi-Layer Perceptrons (MLP). RFs report their own un- certainty estimates through their ensemble, and for GBMs we use heteroscedastic noise on training residuals (std(y− ˆy)). We take two holdout sets in 2018 and 2022, both unprecedented years for heat exposure in the UK. We detrend for long-term trend, seasonality, day-of- week effects, and public holidays using linear and Fourier terms. The outcome y it is a Pearson mortality residual; these are considered our ‘excess deaths’ not covered by these simple detrending terms, including heat-related mortality. We then normalise units to excess deaths per 100k population. For each holdout year and model we report RMSE, MAE and Pearson-correlation, as well as CRPS and Coverage@90% where uncertainty is available. Hyper-parameters for each model are optimised through a Bayesian search over RMSE with 2017 as validation year, though as is standard practise in environmental epidemiology we use the DLNM parameters from Lo et al. [11], a government-cited study on heat risk in the UK. All three RGNN variants substantially outperform DLNM on RMSE in both holdout years. In the Supplementary we provide per-region results and exam- ple risk curves from the vanilla DLNM alongside the optimised curves from our RGNNs. The most striking result is the behaviour of predictive coverage under distributional shift. In the moderate 2018 holdout, all models with uncertainty estimates achieve reasonable coverage. In 2022, however, the DLNM’s coverage 6Davies et al. Fig. 3. RMSE (↓), CRPS (↓), Correlation (↑) and Coverage@90% for the holdout years 2018 (top) and 2022 (bottom). collapses to well below the nominal 90% level—while the RGNN variants main- tain coverage between 0.83 and 0.89. For a public health tool intended to support decision-making during extreme heat events, calibration under tail conditions is arguably more consequential than average-year point accuracy. 4.1 Counterfactuals As an early set of results, we vary census features at OA and examine the changes in predictions made using the RGNN-OA. We perform two experiments here; in- creasing the proportion of OA population with low educational qualifications (level 1/2), as well as the proportion of young adults (25-29), by one standard deviation upwards. We perform this analysis for the London region using tem- perature data for the 21st July 2022, the hottest day on-record in the UK. Chloropleth visualisation in Figure 4 shows heterogeneity across spatial areas, and a sweep over delta magnitudes in Figure 5 (Appendix) shows the expected relationships. Increased proportions of population with low educational levels increases mortality, and increased proportions of young people decrease it, in- line with epidemiological literature on deprivation and clinical risks [12]. The North East in-general shows greater sensitivity to demographic changes, par- ticularly in education levels, possibly due to the broadly lower socio-economic status of the region in-comparison to reason. This spatially-resolved attribution, mapping demographic vulnerability onto urban geography, exemplifies the kind of actionable, area-level intelligence that can directly inform local heat-health interventions. Further analysis will follow in future iterations of this work. 4.2 Discussion The most consequential result is calibration: where DLNM and RF coverage collapse during the 2022 heatwave, all RGNN variants maintain near-nominal Heat-Mortality Risk with GNNs7 Fig. 4.∆ mortality with varying census features for the RGNN-OA model on the 21st July 2022 in the London and North East regions. 8Davies et al. Fig. 5.∆ predictions with varying proportions of young and lower-educated people in OAs, as predicted by the RGNN-OA in London. coverage. For a public health tool intended to support decision-making during the events where it is needed most, this robustness under distributional shift is a meaningful contribution. Early counterfactual experiments provide evidence that the model is learn- ing epidemiologically coherent structure rather than spurious correlations, with deprivation and age effects in the expected directions. More systematic feature attribution analysis will follow in future iterations. A natural extension is to assess whether the learned demographic embeddings transfer across geographic contexts, enabling risk curve estimation for areas with limited mortality data. 5 Conclusion We introduced Risk Graph Neural Networks, which augment classical DLNMs with a hierarchical GNN encoder over census features, preserving interpretable risk curve outputs while substantially improving predictive calibration under extreme heat. Evaluated across 10 regions of England and Wales, RGNN variants maintain near-nominal uncertainty coverage where standard baselines collapse. Future work will broaden the geographic scope, investigate adaptive hierarchy depth, and provide fuller feature attribution analysis. We also intend to explore whether representations learned over England and Wales generalise to other health outcome data. Acknowledgments. Disclosure of Interests. The authors have no competing interests to declare that are relevant to the content of this article. Bibliography [1] J. Boudreault, A. Ruf, C. Campagna, and F. Chebana. Multi-region models built with machine and deep learning for predicting several heat- related health outcomes. Sustainable Cities and Society, 115:105785, Nov. 2024. ISSN 2210-6707. https://doi.org/10.1016/j.scs.2024.105785. URL https://w.sciencedirect.com/science/article/pii/S2210670724006097. [2] J. Boudreault, F. Lamothe, C. Campagna, and F. Chebana. Machine learning for modelling the health impacts of extreme heat: A compre- hensive literature review. Environment International, 206:109965, Dec. 2025. ISSN 0160-4120. https://doi.org/10.1016/j.envint.2025.109965. URL https://w.sciencedirect.com/science/article/pii/S0160412025007160. [3] E. Fillola, R. Santos-Rodriguez, R. Tunnicliffe, J. N. Clark, N. Keshtmand, A. Ganesan, and M. Rigby. Enabling fast greenhouse gas emissions inference from satellites with GATES: a Graph-Neural-Network Atmospheric Trans- port Emulation System. Geoscientific Model Development, 19(5):1893–1915, Mar. 2026. ISSN 1991-959X. https://doi.org/10.5194/gmd-19-1893-2026. URL https://gmd.copernicus.org/articles/19/1893/2026/. [4] A. Gasparrini, B. Armstrong, and M. G. Kenward. Distributed lag non-linear models.Statistics in Medicine, 29(21):2224–2234, 2010. ISSN 1097-0258. https://doi.org/10.1002/sim.3940. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.3940._eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/sim.3940. [5] O. Goodwin and A. Singleton. A unified dataset of UK census variables for small areas: Harmonised data tables from the 2021 England, Wales, and Northern Ireland censuses and the 2022 Scotland census. Environment and Planning B: Urban Analytics and City Science, page 23998083261429563, Feb. 2026. ISSN 2399-8083. https://doi.org/10.1177/23998083261429563. URL https://doi.org/10.1177/23998083261429563. [6] W. L. Hamilton, R. Ying, and J. Leskovec. Representation learning on graphs: Methods and applications. arXiv, 1(1):1–24, Sept. 2017. URL http://arxiv.org/abs/1709.05584. [7] D. Hollis, M. McCarthy, M. Kendon, T. Legg, and I. Simp- son.HadUK-Grid—A new UK dataset of gridded cli- mate observations.Geoscience Data Journal, 6(2):151–159, 2019.ISSN 2049-6060.https://doi.org/10.1002/gdj3.78.URL https://onlinelibrary.wiley.com/doi/abs/10.1002/gdj3.78._eprint: https://rmets.onlinelibrary.wiley.com/doi/pdf/10.1002/gdj3.78. [8] W. Jiang and J. Luo. Graph neural network for traffic forecasting: A survey, Nov. 2022. [9] T. N. Kipf and M. Welling. Semi-Supervised Classification with Graph Con- volutional Networks. In In Proceedings of the 5th International Conference on Learning Representations (ICLR), Apr. 2017. 10Davies et al. [10] L. I.-K. Lin.A Concordance Correlation Coefficient to Evaluate Reproducibility.Biometrics, 45(1):255–268, 1989. ISSN 0006-341X.https://doi.org/10.2307/2532051.URL https://w.jstor.org/stable/2532051. [11] Y. T. E. Lo, D. M. Mitchell, R. Thompson, E. O’Connell, and A. Gas- parrini. Estimating heat-related mortality in near real time for na- tional heatwave plans. Environmental Research Letters, 17(2):024017, Feb. 2022. ISSN 1748-9326. https://doi.org/10.1088/1748-9326/ac4cf4. URL https://doi.org/10.1088/1748-9326/ac4cf4. [12] Y. T. E. Lo, D. M. Mitchell, and A. Gasparrini.Compound mortality impacts from extreme temperatures and the COVID- 19 pandemic.Nature Communications, 15(1):4289, May 2024. ISSN 2041-1723. https://doi.org/10.1038/s41467-024-48207-2. URL https://w.nature.com/articles/s41467-024-48207-2. [13] P. Veličković, G. Cucurull, A. Casanova, A. Romero, P. Liò, and Y. Ben- gio. Graph Attention Networks. In In Proceedings of the 6th In- ternational Conference on Learning Representations (ICLR), Feb. 2018. https://doi.org/10.48550/arXiv.1710.10903. A Appendix Table 2: Per-region holdout results for all models across both hold- out years. RMSE↓, MAE↓, Pearson r ↑, CRPS↓, and Coverage-90 are shown for each region under the 2018 (moderate) and 2022 (se- vere heatwave) holdout years. Bold indicates the best result per metric in each region-year block. For Coverage-90, best is defined as closest to the nominal 0.90 level. The Poisson noise floor (fl) and Pearson ceiling (cl) are shown for reference alongside each re- gion header. † CRPS and Coverage-90 are not reported for LSTM and MLP, as these models do not produce well-defined predictive distributions. Region Model RMSE↓ MAE↓ Pearson↑ CRPS↓Cov-90 North East — 2018 fl = 0.327, cl = 0.630 DLNM0.455 0.346 0.204 0.2650.715 RF0.384 0.286 0.405 0.2510.342 GBM0.360 0.272 0.286 0.2030.808 LSTM † 0.372 0.296 0.540— MLP † 0.413 0.332 0.429— RGNN0.409 0.341 0.266 0.2360.962 RGNN L 0.378 0.306 0.432 0.2200.984 (continued on next page) Heat-Mortality Risk with GNNs11 (continued from previous page) Region Model RMSE↓ MAE↓ Pearson↑ CRPS↓Cov-90 RGNN OA 0.366 0.298 0.426 0.2120.970 North East — 2022 fl = 0.326, cl = 0.631 DLNM0.653 0.521 0.161 0.4200.504 RF0.594 0.482 0.287 0.4450.148 GBM0.594 0.482 0.221 0.3820.506 LSTM † 0.527 0.449 0.240— MLP † 0.654 0.534 0.179— RGNN 0.362 0.260 0.255 0.2020.964 RGNN L 0.393 0.306 0.213 0.2230.975 RGNN OA 0.379 0.285 0.247 0.2110.964 North West — 2018 fl = 0.188, cl = 0.798 DLNM0.321 0.245 0.258 0.1870.737 RF0.287 0.211 0.471 0.1850.351 GBM0.271 0.205 0.303 0.1550.758 LSTM † 0.249 0.194 0.612— MLP † 0.275 0.212 0.362— RGNN0.291 0.242 0.459 0.1710.997 RGNN L 0.228 0.180 0.566 0.1400.997 RGNN OA 0.230 0.183 0.558 0.1390.989 North West — 2022 fl = 0.187, cl = 0.801 DLNM0.587 0.491 0.484 0.4100.307 RF0.582 0.488 0.486 0.4610.071 GBM0.576 0.485 0.494 0.4060.272 LSTM † 0.581 0.523 0.534— MLP † 0.506 0.435 0.554— RGNN 0.336 0.248 0.485 0.1840.959 RGNN L 0.425 0.356 0.397 0.2440.882 RGNN OA 0.397 0.320 0.446 0.2230.893 Yorkshire — 2018 fl = 0.215, cl = 0.743 DLNM0.330 0.242 0.262 0.1890.721 RF0.289 0.212 0.459 0.1850.351 GBM0.275 0.206 0.295 0.1560.783 LSTM † 0.239 0.182 0.618— MLP † 0.270 0.209 0.508— RGNN0.319 0.270 0.403 0.1850.981 RGNN L 0.265 0.213 0.539 0.1570.986 RGNN OA 0.262 0.210 0.521 0.1530.981 Yorkshire — 2022 fl = 0.215, cl = 0.743 DLNM0.558 0.452 0.485 0.3720.414 (continued on next page) 12Davies et al. (continued from previous page) Region Model RMSE↓ MAE↓ Pearson↑ CRPS↓Cov-90 RF0.536 0.436 0.433 0.4080.115 GBM0.532 0.435 0.447 0.3580.369 LSTM † 0.545 0.467 0.435— MLP † 0.535 0.456 0.437— RGNN 0.312 0.235 0.442 0.1750.953 RGNN L 0.375 0.309 0.347 0.2130.934 RGNN OA 0.351 0.278 0.404 0.1960.923 East Midlands — 2018 fl = 0.221, cl = 0.716 DLNM0.335 0.262 0.346 0.1990.710 RF0.290 0.218 0.425 0.1900.392 GBM0.283 0.212 0.335 0.1620.742 LSTM † 0.240 0.188 0.618— MLP † 0.279 0.221 0.526— RGNN0.256 0.210 0.503 0.1500.986 RGNN L 0.226 0.184 0.551 0.1360.989 RGNN OA 0.225 0.182 0.545 0.1330.986 East Midlands — 2022 fl = 0.222, cl = 0.718 DLNM0.548 0.439 0.561 0.3590.449 RF0.535 0.432 0.454 0.4020.101 GBM0.537 0.433 0.340 0.3530.386 LSTM † 0.553 0.469 0.358— MLP † 0.539 0.457 0.482— RGNN 0.339 0.254 0.495 0.1840.945 RGNN L 0.415 0.342 0.396 0.2370.866 RGNN OA 0.389 0.310 0.449 0.2170.868 West Midlands — 2018 fl = 0.202, cl = 0.751 DLNM0.327 0.251 0.353 0.1940.699 RF0.291 0.218 0.529 0.1920.345 GBM0.288 0.215 0.426 0.1650.764 LSTM † 0.253 0.205 0.683— MLP † 0.247 0.193 0.555— RGNN0.238 0.193 0.519 0.1441.000 RGNN L 0.214 0.173 0.562 0.1331.000 RGNN OA 0.210 0.169 0.557 0.1301.000 West Midlands — 2022 fl = 0.202, cl = 0.751 DLNM0.554 0.457 0.533 0.3760.375 RF0.541 0.450 0.530 0.4200.068 GBM0.539 0.447 0.469 0.3690.325 LSTM † 0.559 0.497 0.460— (continued on next page) Heat-Mortality Risk with GNNs13 (continued from previous page) Region Model RMSE↓ MAE↓ Pearson↑ CRPS↓Cov-90 MLP † 0.481 0.405 0.527— RGNN 0.330 0.239 0.508 0.1800.948 RGNN L 0.397 0.325 0.433 0.2240.904 RGNN OA 0.383 0.304 0.471 0.2130.901 East of England — 2018 fl = 0.191, cl = 0.753 DLNM0.319 0.238 0.348 0.1820.742 RF0.285 0.205 0.407 0.1790.378 GBM0.265 0.194 0.395 0.1470.769 LSTM † 0.281 0.216 0.494— MLP † 0.261 0.199 0.508— RGNN 0.215 0.164 0.579 0.1260.973 RGNN L 0.220 0.172 0.597 0.1300.973 RGNN OA 0.220 0.168 0.590 0.1270.964 East of England — 2022 fl = 0.191, cl = 0.756 DLNM0.445 0.359 0.571 0.2900.496 RF0.438 0.354 0.476 0.3260.153 GBM0.442 0.355 0.327 0.2880.447 LSTM † 0.450 0.388 0.365— MLP † 0.412 0.358 0.383— RGNN 0.327 0.269 0.407 0.1860.901 RGNN L 0.398 0.342 0.324 0.2340.827 RGNN OA 0.383 0.325 0.373 0.2240.816 London — 2018 fl = 0.135, cl = 0.801 DLNM0.201 0.159 0.375 0.1190.764 RF0.182 0.142 0.454 0.1230.367 GBM0.173 0.136 0.427 0.1010.797 LSTM † 0.200 0.165 0.442— MLP † 0.183 0.145 0.525— RGNN0.180 0.150 0.633 0.1141.000 RGNN L 0.225 0.192 0.625 0.1361.000 RGNN OA 0.190 0.160 0.629 0.1181.000 London — 2022 fl = 0.134, cl = 0.819 DLNM0.352 0.299 0.450 0.2430.392 RF0.347 0.301 0.459 0.2790.066 GBM0.343 0.294 0.363 0.2390.350 LSTM † 0.351 0.320 0.466— MLP † 0.334 0.299 0.599— RGNN0.366 0.328 0.471 0.2180.901 RGNN L 0.420 0.381 0.448 0.2540.847 (continued on next page) 14Davies et al. (continued from previous page) Region Model RMSE↓ MAE↓ Pearson↑ CRPS↓Cov-90 RGNN OA 0.379 0.341 0.468 0.2270.874 South East — 2018 fl = 0.157, cl = 0.807 DLNM0.272 0.205 0.315 0.1560.770 RF0.246 0.181 0.383 0.1560.359 GBM0.232 0.172 0.370 0.1300.806 LSTM † 0.275 0.199 0.300— MLP † 0.252 0.200 0.442— RGNN 0.198 0.154 0.519 0.1190.986 RGNN L 0.202 0.161 0.530 0.1210.984 RGNN OA 0.203 0.159 0.526 0.1190.975 South East — 2022 fl = 0.157, cl = 0.808 DLNM0.466 0.382 0.569 0.3140.419 RF0.464 0.380 0.462 0.3540.126 GBM0.465 0.381 0.430 0.3160.356 LSTM † 0.474 0.414 0.426— MLP † 0.449 0.393 0.510— RGNN 0.340 0.282 0.442 0.1940.888 RGNN L 0.407 0.351 0.394 0.2420.792 RGNN OA 0.412 0.354 0.418 0.2470.759 South West — 2018 fl = 0.213, cl = 0.740 DLNM0.316 0.239 0.319 0.1800.770 RF0.279 0.205 0.469 0.1770.375 GBM0.263 0.195 0.439 0.1470.833 LSTM † 0.266 0.193 0.518— MLP † 0.258 0.191 0.510— RGNN0.244 0.196 0.543 0.1410.970 RGNN L 0.231 0.178 0.580 0.1340.984 RGNN OA 0.220 0.170 0.565 0.1270.981 South West — 2022 fl = 0.214, cl = 0.741 DLNM0.570 0.462 0.544 0.3780.430 RF0.557 0.456 0.468 0.4260.090 GBM0.560 0.460 0.394 0.3770.361 LSTM † 0.558 0.487 0.448— MLP † 0.509 0.439 0.511— RGNN 0.379 0.294 0.524 0.2090.866 RGNN L 0.521 0.447 0.467 0.3230.630 RGNN OA 0.441 0.362 0.505 0.2590.745 Wales — 2018 fl = 0.303, cl = 0.651 DLNM0.451 0.345 0.376 0.2650.701 (continued on next page) Heat-Mortality Risk with GNNs15 (continued from previous page) Region Model RMSE↓ MAE↓ Pearson↑ CRPS↓Cov-90 RF0.386 0.291 0.474 0.2570.321 GBM0.380 0.288 0.411 0.2190.756 LSTM † 0.340 0.271 0.611— MLP † 0.361 0.295 0.615— RGNN0.310 0.246 0.534 0.1790.973 RGNN L 0.272 0.216 0.616 0.1600.989 RGNN OA 0.279 0.219 0.610 0.1620.981 Wales — 2022 fl = 0.302, cl = 0.651 DLNM0.637 0.504 0.502 0.4050.496 RF0.612 0.483 0.543 0.4470.142 GBM0.607 0.480 0.490 0.3850.461 LSTM † 0.610 0.546 0.475— MLP † 0.619 0.545 0.488— RGNN 0.412 0.302 0.499 0.2190.926 RGNN L 0.486 0.399 0.413 0.2770.868 RGNN OA 0.456 0.358 0.468 0.2520.882 fl = Poisson RMSE noise floor; cl = Pearson ceiling. RGNN L = RGNN-LSOA; RGNN OA = RGNN-OA. 16Davies et al. Fig. 6. Vanilla DLNM curves for each UK region in our experiments. Important to note in contrast with the optimised curves below is the flat portion at moderate tem- peratures, with a ‘bump’ due to the combination of spline specification and heat risk falling below detectable SNR levels. Heat-Mortality Risk with GNNs17 Fig. 7. DLNM curves for each UK region in our experiments, optimised by RGNN- LAD. 18Davies et al. Fig. 8. DLNM curves for each UK region in our experiments, optimised by RGNN- LSOA. Heat-Mortality Risk with GNNs19 Fig. 9. DLNM curves for each UK region in our experiments, optimised by RGNN-OA.